Conference rusure::math

102.0. "About the Polysnake" by XENON::GAUDREAU () Fri Jul 27 1984 16:45

   Here's a tough one written by Martin Gardner in Scientific American, June
1981, pgs 24-29.  Kim's full name is Scott Kim.  Enjoy -


   "First  we  must  define  a  snake.  It  is  a  single connected chain of
identical  unit  cubes  joined  at  their faces in such a way that each cube
(except  for  a  cube  at  the end of the chain) is attached face to face to
exactly  two  other  cubes.  The  Snake may twist in any possible direction,
provided  no  internal  cube  abuts  the face of any cube other than its two
immediate neighbors. The snake may, however, twist so that any number of its
cubes  touch  along  edges  or at corners. A polycube snake may be finite in
length,  having two end cubes that are each fastened to only one cube, or it
may  be finite and closed so that it has no ends. A snake may also have just
one  end and may be infinite in length, or it may be infinite and endless in
both directions.

   We  now ask a deceptively simple question. What is the smallest number of
snakes needed to fill all space? We can put it another way. Imagine space to
be  completely  packed  with  an  infinite number of unit cubes. What is the
smallest  number  of  snakes into which it can be dissected by cutting along
the planes that define the cubes?

   If  we  consider the two-dimensional analogue of the problem (snakes made
of  unit  squares),  it  is  easy  to  see that the answer is two. We simply
intertwine  two  spirals  of  infinite  one-ended flat snakes, one gray, one
white.

   The  question of how to fill three-dimensional space with polycube snakes
is  not  so easily answered. Kim has found a way of twisting four infinitely
long  one-ended  snakes  (it  is convenient to think of them as being each a
different  color)  into  a structure of interlocked helical shapes that fill
all  space. The method is too complicated to explain in a limited space; you
will have to take my word that it can be done.

   Can  it be done with three snakes? Not only is this an unanwered question
but  also  Kim  has been unable to prove that it cannot be done with two! '"A
solution  with only two snakes,"' he wrote in a letter, '"would constitute a
sort of infinite yin-yang: the negative space left by one snake would be the
other  snake.  It is the beauty of such an entwining, and the possibility of
building  a model large enough to crawl through, that keeps me searching for
a solution."'

   The  problem can of course be generalized to snakes made of unit cubes in
any  number  of  dimensions.  Kim  has  conjectured  that  in  a  space of n
dimensions  the  minimum number of snakes that completely fill it is 2(n-1),
but the guess is still a shaky one.

   A  few  years  ago  I  had  the pleasure of explaining the polycube-snake
problem to John Horton Conway, the Cambridge mathematician. When I concluded
by   saying   Kim  had  not  yet  shown  that  two  snakes  could  not  tile
three-dimensional  space, Conway instantly said, '"But it's obvious that--"'
He  checked himself in mid-sentence, stared into three-space for a minute or
two, then exclaimed, '"It's not obvious!"'

   I  have  no idea what passed through Conways mind. I can only say that if
the  impossibility  of filling three-space with two snakes is not obvious to
Conway or to Kim, it probably is not obvious to anyone else."
------------------------

   Okay, this is it.  Any takers??  I'm just totally boggled and I think
that this should amuse somebody...


 Joe
 -=-